Search & Filtering
Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.
(i) 𝑥2 + 𝑦2 + 𝑧2 + 4𝑥 + 3𝑦 − 𝑧 = 0
(ii) 2𝑥2 − 𝑦2 − 𝑧2 + 𝑥𝑦 + 𝑦𝑧 − 𝑧𝑥 = 1
(iii) 𝑥2 + 𝑦2 − 𝑧2 − 2𝑥𝑦 − 3𝑦𝑧 − 6𝑧𝑥 + 𝑥 − 2𝑦 + 5𝑧 + 4 = 0
Show that the conicoid 2𝑥2 + 2𝑦2 + 𝑥𝑦 − 𝑦𝑧 + 𝑧𝑥 + 2𝑥 − 𝑦 + 5𝑧 + 1 = 0 is
central. Hence find its centre.
Transform the equation 𝑥2 + 2𝑦2 − 6𝑧2 − 2𝑥 − 8𝑦 + 3 = 0 by shifting the
origin to (1, 2, 0) without changing the directions of the coordinate axes. What
object does this new equation represent? Give a rough sketch of it.
Show that the perpendiculars drawn from the origin to tangent planes to the cone 𝑥2 − 𝑦2 + 5z2 + 4𝑥𝑦 = 0 lie on the cone 𝑥2 − 𝑦2 + 𝑧2 + 4𝑥𝑦 = 0.
Find the equation of the cylinder with base 𝑥2 + 𝑦2 + 𝑧2 − 3𝑥 − 6𝑧 + 9 = 0, 𝑥 − 2𝑦 + 2𝑧 − 6 = 0.
Find the angle between the lines of intersection of the cone 4𝑥2 + 𝑦2 + 4𝑧2 + 4𝑦𝑧 + 2𝑧𝑥 = 0 and the plane 𝑥 + 2𝑦 + 3𝑧 = 0.
Find the equation of the sphere touching the plane 8𝑥 + 5𝑦 + 3𝑧 + 1 = 0 at (3, −1, −1) and cutting the sphere 𝑥2 + 𝑦2 + 𝑧2 − 2𝑥 + 𝑦 − 𝑧 − 6 = 0 orthogonally.
Show that the plane 2𝑥 + 𝑦 + 2𝑧 = 0 is a tangent plane to the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 2𝑥 + 2𝑦 − 2𝑧 + 2 = 0.
Find the distance of the origin from the plane which passes through (2, 1, 8) , (1, 0, 2) and (−3, 4, 6)
Find the equation of the plane which passes through the line of intersection of the planes 3𝑥 + 4𝑦 − 5𝑧 = 9 and 2𝑥 + 6𝑦 + 6𝑧 = 7 and which is perpendicular to the plane 3𝑥 + 2𝑦 − 5𝑧 + 6 = 0
#339914 on Dec 2023